Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides

Theorem

Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle.

In the words of Euclid:

If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.

(The Elements: Book $\text{I}$: Proposition $21$)

Corollary

The angle between the two line segments from the endpoints of one side to a point inside the triangle is greater than the angle between the other two sides of the triangle.

Proof

Given a triangle $ABC$ and a point $D$ inside it.

We can construct lines connecting $A$ and $B$ to $D$, and then extend the line $AD$ to a point $E$ on $BC$.

In $\triangle ABE$, $AB + AE>BE$.

Then, $AB + AC = AB + AE + EC > BE + EC$ by Euclid's second common notion.

Similarly, $CE + ED > CD$, so $CE + EB = CE + ED + DB > CD + DB$.

Thus, $AB + AC > BE + EC > CD + DB$.

$\blacksquare$

Historical Note

This proof is Proposition $21$ of Book $\text{I}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions